I Prefer That You Wait

Alice and Betty agree to meet tomorrow between 12:00 and 13:00. Alice will arrive uniformly at random during this period, and will wait for 15 minutes or 13:00, whichever comes first. Betty is more patient, and will wait for 20 minutes or 13:00, whichever comes first.

The probability that Alice and Betty will meet each other can be written as \( \dfrac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. Find \( a + b \).

Note.- Alice and Betty's arrival times are independent of each other,

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